Part of my worldview is that progress, innovation and competence in all areas of science, technology, and other aspects of civilization are correlated.

I'm sure they're correlated but not all that tightly.

What would the world look like if that hypothesis were false? Well, we could find a country that is not particularly competent overall, but was very competent and innovative in one specific civilizational subfield. As a random example, imagine it turned out that Egypt actually had the world's best research and technology in the field of microbiology.

I think there are some pretty good examples. The soviets made great achievements in spaceflight and nuclear energy research in spite of having terrible economic and social policies. The Mayans had sophisticated astronomical calendars but they also practiced human sacrifice and never invented the wheel.

If the theory is true, then the fact that the US still seems innovative in CS-related fields is probably a transient anomaly.

I doubt it, but even if true it doesn't save us, since plenty of other countries could develop AGI.